Abstract

We calculate trapping forces, trap stiffness and interference effects for spherical particles in optical tweezers using electromagnetic theory. We show the dependence of these on relative refractive index and particle size. We investigate resonance effects, especially in high refractive index particles where interference effects are expected to be strongest. We also show how these simulations can be used to assist in the optimal design of traps.

Figures (10)

Axial (a) and radial (b) forces acting on a particle in optical tweezers. Here, the radius of the particle is 1λ, with a relative refractive index of nrel = 1.45/1.33 = 1.09, trapped using an objective with numerical aperture of 1.25. The two most important features of the trapping force are shown: the maximum reverse axial force (A) which characterizes the strength of the trap, and the radial spring constant (the slope of line B).

Axial restoring forces (Q) for (a) NA = 1.0, (b) NA = 1.2 and (c) NA = 1.3 corresponding to beam convergence angles of 49°, 64° and 78° in water. The strongest restoring forces are marked with points and scale non-linearly with numerical aperture. The effect of convergence angle and interference on the restoring force is such that the parameter space allows trapping of a greater range of particles. Particles typically thought of as untrappable also become available for use in optical tweezers. Though visibility is smaller in the strong trapping regions the effect of interference can be seen.

(a) Maximum trap strength force as a function of numerical aperture and (b) relative refractive index and particle size at which the maximum trap strength occurs. The maximum trap strengths shown in (a) are the highest values that occur in trapping landscapes such as those shown in Fig. 2; the locations of these maximum values are shown in (b). The numerical aperture of the focusing objective lens has a non-linear relationship to restoring force. The particle size and relative refractive index of the maximum of restoring force were found show a quadratic dependence, with nrel = 0.4137r2-2.5005r+4.7442.

Dependence of trap strength force for (a) polystyrene and (b) diamond on particle size. Interference effects in the larger particles are clearly visible in these graphs. The period of interference is not simply that which would result from the usual thin-film interference effects (λ/2), as the particle is not planar, and the illuminating wave is not plane. The interference effects in diamond have a larger amplitude than polystyrene due to the large difference in refractive index between the medium and particle.

Momentum of light field for two similarly sized particles (particle radius of 0.947λ and 0.985λ) showing greatly different scattering properties. The trapped particle (red/dark gray) causes very little backscatter in comparison to the particle 0.076λ in radius larger. The radiation pattern from the larger particle (cyan/gray) has a significant backscattered component which is generated from multiple passes of the light within its structure. Note that the difference between the two radii is small—the reflectivity of a planar dielectric sheet will not change by as much for the same change in thickness.

(a) Mie interference structure overlayed with trapping force for a nrel = 1.82 particle and (b) detailed plot of extinction efficiency and trapping force within a particle radius range of 0.8λ to 1λ. The large scale structure of the extinction efficiency does not affect trapping, as this is due to interference between light passing through the particle and light that misses the particle—in optical tweezers, all of the light typically passes through the particle. Particular modes in the Mie interference structure match those in a trapped particle; when this occurs we see coincident peaks in the trapping force and the extinction curves.

Effect of absorption on trap strength. The real part of the relative refractive index is nrel = 1.82 and the imaginary parts are (a) nim = 0.0001, (b) nim = 0.001, (c) nim = 0.01 and (d) nim = 0.1. Here we can see that the location of peaks in the plot of force remain unchanged. However, as the imaginary component of refractive index is increased, trapping outside the Rayleigh range becomes impossible.

Radial spring constants (i.e., the trap stiffness) for (a) NA = 1.0, (b) NA = 1.2 and (c) NA = 1.3 corresponding to beam convergence angles of 49°, 64° and 78° in water. As a trap is only viable when there is a restoring force acting upon the particle, we only show trap stiffnesses for particles which are also axially trapped. However, there is still a periodic relationship with the particle radius.

Force profiles for three standard sized polystyrene spheres. The gradient of the force profiles changes most for large particles, but we can see this effect here in more standard sized particles found in optical tweezers (particles on the order of an optical wavelength). Especially for the force profile of the largest particle, we find that there is an axial position dependence on the trap stiffness for displacements near the equilibrium.

Plots of equilibrium position with radial position for 1λ particles of nrel = 1.1 (red), nrel = 1.25 (green) and nrel = 1.5 (blue). In the white regions, the axial force is in the direction of beam propagation, and in the colored regions, in the opposite direction. The axial equilibrium position at a particular radial displacement lies on the upper boundary of the colored regions. In the high refractive index case the particle can only be trapped if it is pushed into the trap from above the beam focus as the attractive component of particle position is enclosed by the repulsive region. For the other two curves the particle may be trapped from below the beam focus.